Answer :
To solve this problem, we are going to use the concept of normal distribution. A normal distribution is a bell-shaped distribution that is symmetric about the mean. We'll use the mean and standard deviation to find the necessary values.
### Part C: Proportion of healthy adults with body temperatures below 97.8 degrees
1. Understand the Parameters:
- We are assuming the average body temperature (mean) of healthy adults is 98.6 degrees Fahrenheit.
- The standard deviation for body temperature is assumed to be 0.7 degrees.
2. Find the Z-score:
- The Z-score is a measure of how many standard deviations a data point is from the mean. The formula for the Z-score is:
[tex]\[
Z = \frac{(X - \text{mean})}{\text{standard deviation}}
\][/tex]
- Here, [tex]\( X \)[/tex] is 97.8 degrees.
- So, calculate the Z-score:
[tex]\[
Z = \frac{(97.8 - 98.6)}{0.7} = -1.14
\][/tex]
3. Find the Proportion:
- Use the Z-score to find the cumulative probability from the standard normal distribution table (or tool). This gives us the proportion of the population with a body temperature below 97.8 degrees.
- The result for this is approximately 0.13, meaning 13% of the population.
### Part D: Body temperature for the top 1% of healthy adults
1. Find the Z-score for the Top 1%:
- We want to find the body temperature at which only 1% of the population is above it. This refers to the 99th percentile.
- The Z-score corresponding to the 99th percentile is approximately 2.33.
2. Calculate the required body temperature:
- Use the Z-score to find the X value (body temperature) using the formula:
[tex]\[
X = \text{mean} + (Z \times \text{standard deviation})
\][/tex]
- Substitute the values:
[tex]\[
X = 98.6 + (2.33 \times 0.7) = 100.23
\][/tex]
3. Result:
- Therefore, the top 1% of healthy adults would have a body temperature of at least 100.23 degrees Fahrenheit.
In conclusion, 13% of healthy adults have body temperatures below 97.8 degrees, and the body temperature at least 100.23 degrees would place someone in the top 1% of healthy adults.
### Part C: Proportion of healthy adults with body temperatures below 97.8 degrees
1. Understand the Parameters:
- We are assuming the average body temperature (mean) of healthy adults is 98.6 degrees Fahrenheit.
- The standard deviation for body temperature is assumed to be 0.7 degrees.
2. Find the Z-score:
- The Z-score is a measure of how many standard deviations a data point is from the mean. The formula for the Z-score is:
[tex]\[
Z = \frac{(X - \text{mean})}{\text{standard deviation}}
\][/tex]
- Here, [tex]\( X \)[/tex] is 97.8 degrees.
- So, calculate the Z-score:
[tex]\[
Z = \frac{(97.8 - 98.6)}{0.7} = -1.14
\][/tex]
3. Find the Proportion:
- Use the Z-score to find the cumulative probability from the standard normal distribution table (or tool). This gives us the proportion of the population with a body temperature below 97.8 degrees.
- The result for this is approximately 0.13, meaning 13% of the population.
### Part D: Body temperature for the top 1% of healthy adults
1. Find the Z-score for the Top 1%:
- We want to find the body temperature at which only 1% of the population is above it. This refers to the 99th percentile.
- The Z-score corresponding to the 99th percentile is approximately 2.33.
2. Calculate the required body temperature:
- Use the Z-score to find the X value (body temperature) using the formula:
[tex]\[
X = \text{mean} + (Z \times \text{standard deviation})
\][/tex]
- Substitute the values:
[tex]\[
X = 98.6 + (2.33 \times 0.7) = 100.23
\][/tex]
3. Result:
- Therefore, the top 1% of healthy adults would have a body temperature of at least 100.23 degrees Fahrenheit.
In conclusion, 13% of healthy adults have body temperatures below 97.8 degrees, and the body temperature at least 100.23 degrees would place someone in the top 1% of healthy adults.